Legally, How tied in with Putin is Trump?

The only way really to know likely is through Trump's Tax returns. Trump is the first president since Nixon started releasing his tax returns to hide his tax returns from the public which is extremely suspicious in the first place.
begin partial quote from:

https://www.google.com/search?q=which+presidents+made+public+their+tax+returns%3F&rlz=1C5CHFA_enUS750US750&oq=which+presidents+made+public+their+tax+retur

When did presidential candidates start releasing tax returns?

In 1973, while under audit and amid a controversy over his personal taxes, Nixon released returns dating back to when he took office in 1969. The scandal led to one of Nixon's most famous quotes. "People have got to know whether or not their president is a crook," Nixon told reporters in November 1973.Jan 26, 2017

So, the only way to really know whether Trump is tied into Putin more than we already suspect is

to see his tax returns, especially 2016 and 2017 and possibly ones before this too.